34. MULTIPLES.

Name a number that is 3 times 9.

tiple of 9; it is also a multiple of 3.

27 is a Mul

A multiple of 3 is the product of 3 multiplied by any whole number. A multiple of 9 is any number of times 9. 24, 21, and 28 are multiples of what numbers?

35. A Multiple of a Number is any integral (or whole) number of times that number.

Find the multiples which 3 and 5 have in common. Of the common multiples 15, 30, 45, and 60, which is the least?

36. A multiple that is common to two or more numbers is called a Common Multiple, and the least of the common multiples is called the Least Common Multiple (L. C. M.).

37. Find the L. C. M. of 27, 24, and 30.

2 x 2 x 2 x 3 x 3 x 3 x 5 1080, L. C. M.

=

Resolve 27, 24, and 30 into their prime factors. Select the different factors and repeat each as many times as it is found in any one of the numbers. The product of 2, 2, 2, 3, 3, 3, and 5 is the L. C. M. of 27, 24, and 30.

A multiple of a number must contain at least all the prime factors of that number.

The least common multiple of two or more numbers must contain all the prime factors that enter into each of them, and no others.

Find the quotient of 4 × 6 divided by 2 × 6. Of 4 × 8 divided by 2 × 8.

(4 × 7) ÷ (2 × 7) = ?

(6 × 8) ÷ (3 × 8) = ?

2 times any number is contained in 4 times that number, how many times?

(6 x 7) (3 × 7) = ?

?.

(9 × 5) ÷ (3 × 5) = ?

3 x 9 is contained in 6 x 9 as many times as 3 is contained in what? 3 times any number is contained how many times in 6 times that number?

39. To find quotients by the shortest method, strike out equal factors from the dividend and the divisor.

divisor by the same

Dividing both dividend and

number does not affect the quotient.

The process of striking out equal factors from the dividend and the divisor is called Cancellation.

40. Divide 18 × 5 × 6 by 7 × 4 × 5.

Cancel the common factor 5 from both dividend and divi

9

3

18 × 5 × 6

27

=

=

39

7X

2

sor, and the common factor 2 from 6 in the dividend and 4 in the divisor. Cancel the remaining factor 2 from the 4 in the divisor with the factor 2 in 18 in the dividend. The

product of the factors remaining in the dividend is 27, and of those in the divisor, 7. 27 divided by 7 gives the quotient 34.

Divide 9 × 2 × 6 by 6 × 6 × 3.

NOTE. When the divisor equals the dividend, the quotient is 1. This result will be found when all the factors of the dividend and the divisor are canceled.

1. How many pounds of sugar at 6 cents a pound can be bought for 9 yards of calico at 12 cents a yard?

9 × 12 6

=

9 × 12 cents equals the cost of the 18 calico. As many pounds of sugar must be given for the calico as there

are 6 cents in 9 x 12 cents.

2. How many pairs of shoes at $3.00 a pair must be given in exchange for 30 bushels of potatoes at 50 cents a bushel ?

3. A farmer sold to a grocer 19 bushels of apples at 75 cents a bushel, and took his pay in coffee at 30 cents a pound. How many pounds did he receive?

4. A milkman sells daily 50 quarts of milk at 6 cents a quart. How many yards of carpet at 90 cents a yard can be bought for the milk sold in 7 days?

5. For 12 days' work of 8 hours each a workman received $24. How much at that rate would he receive for 18 days' work of 9 hours each?

6. Mr. Jones sold 121 pounds of beef at 14 cents a pound, and took his pay in potatoes at 77 cents a bushel; how many bushels did he receive?

7. On each of 6 days a grocer sold 150 pounds of sugar at 8 cents a pound. How many pounds of tea must he sell on each of 4 days at 60 cents a pound to equal the amount he received for the sugar?

8. Make a problem for 24 divided by 3 × 2.

9. Make a problem for 45 divided by 3 x 5.

parts of a unit.

One of the equal parts into which the unit is divided is a fractional unit. The value of a fraction depends both upon the size of the fractional unit and the number of fractional units expressed.

many thirds in an

How many halves in a dollar? an apple? a bushel ? a mile ? How orange ? a loaf of bread? in a peck? in a quart?

How many sevenths

2. A fraction is expressed by two numbers, one written under the other, with a horizontal line between them; as, 3, 1, 3, etc.

The number below the line shows the number of equal parts into which the unit is divided; it is called the Denominator.